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Derivative of Logarithmic Functions Calculator

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1

Here, we show you a step-by-step solved example of derivative of logarithmic functions. This solution was automatically generated by our smart calculator:

$\int_1^2x\ln\left(x\right)dx$
2

We can solve the integral $\int x\ln\left(x\right)dx$ by applying integration by parts method to calculate the integral of the product of two functions, using the following formula

$\displaystyle\int u\cdot dv=u\cdot v-\int v \cdot du$

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{x}$
3

First, identify or choose $u$ and calculate it's derivative, $du$

$\begin{matrix}\displaystyle{u=\ln\left(x\right)}\\ \displaystyle{du=\frac{1}{x}dx}\end{matrix}$
4

Now, identify $dv$ and calculate $v$

$\begin{matrix}\displaystyle{dv=xdx}\\ \displaystyle{\int dv=\int xdx}\end{matrix}$
5

Solve the integral to find $v$

$v=\int xdx$
6

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{2}x^2$

The integral of a constant times a function is equal to the constant multiplied by the integral of the function

$\frac{1}{2}x^2\ln\left|x\right|-\frac{1}{2}\int_{1}^{2}\frac{1}{x}x^2dx$

Multiplying the fraction by $x^2$

$\frac{1}{2}x^2\ln\left|x\right|-\frac{1}{2}\int_{1}^{2}\frac{x^2}{x}dx$

Simplify the fraction $\frac{x^2}{x}$ by $x$

$\frac{1}{2}x^2\ln\left|x\right|-\frac{1}{2}\int_{1}^{2} xdx$
7

Now replace the values of $u$, $du$ and $v$ in the last formula

$\left[\frac{1}{2}x^2\ln\left|x\right|\right]_{1}^{2}-\frac{1}{2}\int_{1}^{2} xdx$

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$-\frac{1}{2}\left[\frac{1}{2}x^2\right]_{1}^{2}$

Evaluate the definite integral

$-\frac{1}{2}\cdot \left(\frac{1}{2}\cdot 2^2-\frac{1}{2}\cdot 1^2\right)$

Simplify the expression inside the integral

$-\frac{3}{4}$
8

The integral $-\frac{1}{2}\int_{1}^{2} xdx$ results in: $-\frac{3}{4}$

$-\frac{3}{4}$
9

Gather the results of all integrals

$\left[\frac{1}{2}x^2\ln\left|x\right|\right]_{1}^{2}-\frac{3}{4}$
10

Evaluate the definite integral

$\frac{1}{2}\cdot 2^2\ln\left|2\right|- \frac{1}{2}\cdot 1^2\ln\left|1\right|-\frac{3}{4}$

Calculate the power $2^2$

$\frac{1}{2}\cdot 4\ln\left|2\right|- \frac{1}{2}\cdot 1^2\ln\left|1\right|-\frac{3}{4}$

Multiply $\frac{1}{2}$ times $4$

$2\ln\left|2\right|- \frac{1}{2}\cdot 1^2\ln\left|1\right|-\frac{3}{4}$

Calculate the power $1^2$

$2\ln\left|2\right|- \frac{1}{2}\ln\left|1\right|-\frac{3}{4}$

Calculating the natural logarithm of $2$

$2\ln\left(2\right)- \frac{1}{2}\ln\left|1\right|-\frac{3}{4}$

Multiply $2$ times $\ln\left(2\right)$

$\ln\left(4\right)- \frac{1}{2}\ln\left|1\right|-\frac{3}{4}$

Calculating the natural logarithm of $1$

$\ln\left(4\right)+0-\frac{3}{4}$

$x+0=x$, where $x$ is any expression

$\ln\left(4\right)-\frac{3}{4}$

Subtract the values $\ln\left(4\right)$ and $-\frac{3}{4}$

$0.6362944$
11

Simplify the expression inside the integral

$0.6362944$

Final answer to the problem

$0.6362944$

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